On a sheet pan, place the squash flesh side down and roast until very tender (about 45 minutes). At the same time, bake potatoes directly on the rack of oven for 1 hour.
Split the potatoes and allow to cool slightly, or until you can handle them. Don't let them cool completely. Scoop the flesh of the potatoes and the squash into a bowl and mash with a hand masher. Mix in the egg, salt and nutmeg. Then add the flour and mix until a soft dough forms. Do not do this in a mixer, it will overwork the dough. Add flour by the spoonful if it's still too moist.
Turn out onto a floured board and divide into 8 portions. Roll out into ropes and cut into 1/2-inch pieces. Line the pieces up on a floured sheet pan as you work. At this point you could freeze them on the pan until solid, then transfer to zip top bags and store in the freezer.
In a large pot of boiling, salted water gently drop in the dumplings. Don't overcrowd. As they begin to float, remove them with a slotted spoon and toss them into an ice bath.
Drain off the water and toss in a little oil. Store loosely in containers until ready to use.
To reheat, in a saute pan over high heat add 1 tablespoon of soft butter. Cook until the butter begins to foam and turn brown. Add 2 teaspoons sage leaves and 1 cup of dumplings. Cook for an additional minute until the dumplings are heated through. Repeat until you have desired amount of servings. Plate and top with freshly grated Parmesan.